.center[Variation in Naive T cell pool size]




.center[Measuring cell division in T lymphocytes]



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.pull-left[ ] .pull-right[
- T cells express Ki67 protein for ~ 4 days after the division. .right[Hogan et. al., PNAS 2015]] - Hard to untangle the origin of Ki67 expression within peripheral naive T cells. Thymic Inheritance or division in periphery?



Mathematical models


.center[Potential mechanisms of naive T cell maintenance]


Model vaildation and comparison


\[ \text{Joint Likelihood} \Rightarrow p(y | \hat \theta) = \prod_{i=1}^n \, \text{Normal}(y_i - \text{model}(\theta)_i^\text{pred}, \, \sigma) \]

??? We form a joint likelihood for each model assuming the errors were normally distributed.

\[ p(\hat \theta | y) = \frac{p(y | \hat \theta) \times p(\theta_\text{prior})} {p(y)} \]


Models are selected based on parsimony and their ability to predict New observations — (out-of-sample prediction error).

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.left[Fits for CD4 data]


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.left[Fits for CD8 data]


.left[Two-compartment models]



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Adaptation: age-structured models




Naive T cells of age ‘a’ and Ki67 expression ‘k’ are tracked over time by analytically solving the 3 dimensional PDE,

.shadedbox[ \[ \begin{aligned} \frac{\partial N}{\partial t} + \frac{\partial N}{\partial a} - \beta \, k \, \frac{\partial N}{\partial k}= -(\rho(a) + \delta(a)) \, N(t, a, k) \end{aligned} \]]


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